Simplify and expand the following expression: $ \dfrac{3y + 2}{y - 4}+\dfrac{y + 7}{y - 10} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(y - 4)(y - 10)$ Multiply the first term by $\dfrac{y - 10}{y - 10}$ $ \begin{align*} \dfrac{3y + 2}{y - 4} \times \dfrac{y - 10}{y - 10} & = \dfrac{(3y + 2)(y - 10)}{(y - 4)(y - 10)} \\ & = \dfrac{3y^2 - 28y - 20}{(y - 4)(y - 10)}\end{align*} $ Multiply the second term by $\dfrac{y - 4}{y - 4}$ $ \begin{align*} \dfrac{y + 7}{y - 10} \times \dfrac{y - 4}{y - 4} & = \dfrac{(y + 7)(y - 4)}{(y - 10)(y - 4)} \\ & = \dfrac{y^2 + 3y - 28}{(y - 10)(y - 4)}\end{align*} $ Now we have: $ = \dfrac{3y^2 - 28y - 20}{(y - 4)(y - 10)} + \dfrac{y^2 + 3y - 28}{(y - 10)(y - 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{3y^2 - 28y - 20 + y^2 + 3y - 28}{(y - 4)(y - 10)} $ $ = \dfrac{4y^2 - 25y - 48}{(y - 4)(y - 10)}$ Expand the denominator: $ = \dfrac{4y^2 - 25y - 48}{y^2 - 14y + 40}$